Regularity theorem for totally nonnegative flag varieties
Pavel Galashin
∗1, Steven N. Karp
†2, and Thomas Lam
‡31 Department of Mathematics, University of California, Los Angeles, CA, USA
2 Laboratoire de combinatoire et d’informatique mathématique, Université du Québec à Mon- tréal, Canada
3 Department of Mathematics, University of Michigan, Ann Arbor, MI , USA
Abstract. We show that the totally nonnegative part of a partial flag variety G/P (in the sense of Lusztig) is a regular CW complex, confirming a conjecture of Williams. In particular, the closure of each positroid cell inside the totally nonnegative Grassman- nian is homeomorphic to a ball, confirming a conjecture of Postnikov.
Keywords: Total positivity, algebraic group, partial flag variety, Richardson varieties, totally nonnegative Grassmannian, positroid cells.
1 Introduction
Let G be a semisimple algebraic group, split over R, and let P ⊂ G be a parabolic sub- group. Lusztig [26] introduced the totally nonnegative part of the partial flag variety G/P, denoted(G/P)≥0, which he called a “remarkable polyhedral subspace”. He con- jectured and Rietsch proved [32] that(G/P)≥0 has a decomposition into open cells. We prove the following conjecture of Williams [40]:
Theorem 1.1. The cell decomposition of(G/P)≥0forms a regular CW complex. Thus the closure of each cell is homeomorphic to a closed ball.
A special case of particular interest is when G/P is the Grassmannian Gr(k,n) of k- dimensional linear subspaces ofRn. In this case,(G/P)≥0becomes thetotally nonnegative GrassmannianGr≥0(k,n), introduced by Postnikov [29] as the subset of Gr(k,n)where all Plücker coordinates are nonnegative. He gave a stratification of Gr≥0(k,n) into positroid cells according to which Plücker coordinates are zero and which are strictly positive, and conjectured that the closure of each positroid cell is homeomorphic to a closed ball.
Postnikov’s conjecture follows as a special case ofTheorem 1.1:
∗[email protected]. Supported by NSF grant DMS–1954121.
†[email protected]. Supported by an NSERC postdoctoral fellowship.
‡[email protected]. Supported by a von Neumann Fellowship from the Institute for Advanced Study and by NSF grant DMS-1464693.
Corollary 1.2. The decomposition ofGr≥0(k,n)into positroid cells forms a regular CW complex.
Thus the closure of each positroid cell is homeomorphic to a closed ball.
When k = 1, Gr≥0(1,n) is the standard (n−1)-dimensional simplex ∆n−1 ⊂ Pn−1, a prototypical example of a regular CW complex.
1.1 History and motivation
A matrix is called totally nonnegative if all its minors are nonnegative. The theory of such matrices originated in the 1930’s [35, 15]. Later, Lusztig [26] was motivated by a question of Kostant to consider connections between totally nonnegative matrices and his theory of canonical bases for quantum groups [25]. This led him to introduce the totally nonnegative part G≥0 of a split semisimple G. Inspired by a result of Whitney [39], he defined G≥0 to be generated by exponentiated Chevalley generators with positive real parameters, and generalized many classical results for G = SLn to this setting.
He introduced the totally nonnegative part (G/P)≥0 of a partial flag variety G/P, and showed [27, Section 4] that G≥0and (G/P)≥0 are contractible.
Fomin and Shapiro [7] realized that Lusztig’s work may be used to address a long- standing problem in poset topology. Namely, the Bruhat order of the Weyl group W of G had been shown to be shellable by Björner and Wachs [4], and by general results of Björner [3] it followed that there exists a “synthetic” regular CW complex whose face poset coincides with(W,≤) (see Figure 1). The motivation of [7] was to answer a natu- ral question due to Bernstein and Björner of whether such a regular CW complex exists
“in nature”. Let U ⊂ G be the unipotent radical of the standard Borel subgroup, and let U≥0 := G≥0∩U be its totally nonnegative part. For G = SLn, U≥0 is the semi- group of upper-triangular unipotent matrices with all minors nonnegative. The work of Lusztig [26] implies thatU≥0 has a cell decomposition whose face poset is(W,≤). The spaceU≥0 is not compact, but Fomin and Shapiro [7] conjectured that taking the link of the identity element in U≥0, which also has (W,≤) as its face poset, gives the desired regular CW complex. Their conjecture was confirmed by Hersh [19]. Hersh’s theorem also follows as a corollary to our proof of Theorem 1.1, seeSection 4.1.
Corollary 1.3([19]). The link of the identity in U≥0 is a regular CW complex.
For recent related developments, see [5].
Meanwhile, Postnikov [29] defined the totally nonnegative Grassmannian Gr≥0(k,n), decomposed it into positroid cells, and showed that each positroid cell is homeomorphic to an open ball. Motivated by work of Fomin and Zelevinsky [8] on double Bruhat cells, he conjectured [29, Conjecture 3.6] that this decomposition forms a regular CW complex.
It was later realized that the space Gr≥0(k,n)and its cell decomposition coincide with the one studied by Lusztig and Rietsch in the special case thatG/P=Gr(k,n). Williams [40, Section 7] extended Postnikov’s conjecture from Gr≥0(k,n) to(G/P)≥0.
id
s1 s2
s2s1 s1s2
w0
w0 s1s2
s2s1 s1 s2
Figure 1: The Bruhat order onW = S3 (after deleting the bottom element) is the face poset of the regular CW complex homeomorphic to a 2-dimensional ball on the right.
There has been much progress towards proving these conjectures. Williams [40]
showed that the face poset of (G/P)≥0 (and hence of Gr≥0(k,n)) is graded, thin, and shellable, and therefore by [3] is the face poset of some regular CW complex. Postnikov, Speyer, and Williams [30] showed that Gr≥0(k,n)is a CW complex, and their result was generalized to (G/P)≥0 by Rietsch and Williams [33]. Rietsch and Williams [34] also showed that the closure of each cell in (G/P)≥0 is contractible. In previous work [11, 13], we showed that the spaces Gr≥0(k,n) and (G/P)≥0 are homeomorphic to closed balls, which is the special case of Theorem 1.1 for the top-dimensional cell of (G/P)≥0. We remark that our proof of Theorem 1.1 uses different methods than those employed in [11, 13], in which we relied on the existence of a vector field on G/P contracting (G/P)≥0 to a point in its interior. Singularities of lower-dimensional positroid cells give obstructions to the existence of a continuous vector field with analogous properties.
Totally positive spaces have attracted a lot of interest due to their appearances in other contexts such as cluster algebras [9] and the physics of scattering amplitudes [1].
Our original motivation for studying the topology of spaces arising in total positivity was to better understand the amplituhedra of Arkani-Hamed and Trnka [2], and more generally the Grassmann polytopes of the third author [22]. A Grassmann polytope is a generalization of a convex polytope in the Grassmannian Gr(k,n). For example, the totally nonnegative Grassmannian Gr≥0(k,n) is a generalization of a simplex, while amplituhedra generalize cyclic polytopes [38]. The faces of a Grassmann polytope are linear projections of closures of positroid cells, and therefore it is essential to understand the topology of these closures in order to develop a theory of Grassmann polytopes.
1.2 Outline
We provide some background definitions in Section 2. We give a brief overview of our proof of Theorem 1.1 in Section 3. Finally, in Section 4 we consider three examples:
the unipotent radicalUn, the complete flag variety Fln, and the Grassmannian Gr(k,n). Further details and full proofs of the results stated here appear in our preprint [12].
2 Background
In this section we review background on regular CW complex and totally nonnegative partial flag varieties. We refer to [16, 24, 26] for further details.
2.1 Regular CW complexes
Let X be a Hausdorff space. We call a finite disjoint union X = Fg∈QXg a regular CW complexif it satisfies the following two properties.
(1) For each g ∈ Q, there exists a homeomorphism from the closure Xg to a closed ball B which sends Xg to the interior of B.
(2) For each g∈ Q, there exists Q0 ⊂Qsuch that Xg =Ff∈Q0Xf.
Theface poset ofX is the poset(Q,), where f g if and only ifXf ⊂Xg.
2.2 Totally nonnegative partial flag varieties
Let g denote the Lie algebra of G over R. We fix Chevalley generators (ei, fi)i∈I ofg, so that the elements hi := [ei, fi] (i∈ I) span the Lie algebra of a split real maximal torus T ofG. Fori∈ I and t ∈R, we define the elements ofG
xi(t) :=exp(tei), yi(t):=exp(t fi).
We also letα∨i : R∗ → T be the homomorphism of algebraic groups whose tangent map takes 1 ∈ R to hi. The xi(t)’s (respectively, yi(t)’s) generate the unipotent radical of a Borel subgroup B(respectively, B−) of G, with B∩B− =T. The data (T,B,B−,xi,yi;i ∈ I) is called apinningfor G.
We define the totally nonnegative part G≥0 of G as the semigroup generated by all xi(t)’s, yi(t)0s, and α∨i (t)’s with t > 0. For a parabolic subgroup P ⊃ B, we define the totally nonnegative part(G/P)≥0 ofG/Pas the closure of the image of G≥0 insideG/P.
For examples in the case G=SLn, see Section 4.
Rietsch [32, 31] established the decomposition (G/P)≥0 = G
g∈Q
Π>g0 (2.1)
of (G/P)≥0 into open balls Π>g0 indexed by the elements g of a certain poset (Q,), which is the face poset of (G/P)≥0. When (G/P)≥0 is the totally nonnegative Grass- mannian Gr≥0(k,n), this is the positroid cell decomposition of [29] (see Section 4.3).
id
s1 s2
s1s2
s2s1
w0
Π>g0 ν¯g
−→
s1 s2s1 Π>g0
w0
s2 id
s1s2
Figure 2: The map ¯νgfor the case G=SL3 andP=BfromExample 4.2.
3 Stars, links, and the Fomin–Shapiro atlas
In this section, we outline our proof ofTheorem 1.1. Given g∈ Q, define thestarof gin (G/P)≥0 by
Star≥g0 := G
hg
Π>h0.
We also consider the space Lk≥g0 (thelinkof g), stratified as Lk≥g0 = G
hg
Lk>g,h0.
We show along the way that Lk≥g0 is a regular CW complex homeomorphic to a closed ball.
At the core of our approach is a collection of (stratification-preserving) homeomor- phisms
¯
νg : Star≥g0 −→∼ Π>g0×Cone(Lk≥g0), (3.1) one for each g ∈ Q (seeFigure 2). Here Cone(A) := (A×R≥0)/(A× {0}) denotes the open cone over A. The homeomorphisms ¯νg, along with dilation actions ϑg on the cones, are part of the data of what we call a Fomin–Shapiro atlas. Our construction is inspired by similar maps introduced in [7] for the unipotent radicalU≥0.
We also introduce the abstract notion of a totally nonnegative space, which captures several known combinatorial and geometric properties of (G/P)≥0 used in our proof.
This includes the shellability ofQdue to Williams [40], and some topological results [31, 21] on Richardson varieties. We prove that every totally nonnegative space that admits a Fomin–Shapiro atlas is a regular CW complex. Our argument proceeds by induction on the dimension of Lk>g,h0, and depends on a delicate interplay between objects in smooth
and topological categories. We use crucially that the maps (3.1) in a Fomin–Shapiro atlas are restrictions of smooth maps. On the topological level, we rely on the generalized Poincaré conjecture [36, 10, 28] combined with some general results on poset topology.
We formulate our results in the abstract language of totally nonnegative spaces since we expect that they can be applied in other contexts, such as to totally nonnegative Kac–
Moody flag varieties, totally nonnegative double Bruhat cells [8], spaces of electrical networks [23], spaces of boundary correlations of planar Ising models [14], amplituhe- dra [2], and the totally nonnegative part of the wonderful compactification [17].
The bulk of the proof of Theorem 1.1 is devoted to the construction of the Fomin–
Shapiro atlas. For each g ∈ Q we give an isomorphism ¯ϕu between an open dense subset Og ⊂ G/P and a certain subset of theaffine flag variety G/B of the loop group G associated with G. The map ¯ϕu, which we call an affine Bruhat atlas, sends the projected Richardson stratification [21] of G/P to the affine Richardson stratification of its image insideG/B. The hardest part of the proof consists of showing that the subsetOg⊂ G/P contains Star≥g0.
Remark 3.1. The map ¯ϕu generalizes the map of Snider [37] from Gr(k,n) to all G/P.
A different approach to give such a generalization is due to He, Knutson, and Lu [18], which led them to the notion of aBruhat atlas. See [6] for the definition. Huang [20] has independently constructed a map similar to our ¯ϕu.
4 Examples
In this section we discuss three examples (in type A) of regular CW complexes which are addressed by Theorem 1.1. We fix n≥ 1, and let[n] :={1, 2, . . . ,n}. For 0 ≤ k ≤n, let ([nk]) denote the set of all k-element subsets of n. We set G := SLn. In the setup of Section 2.2, we may take I := [n−1], wherexi(t), yi(t), andα∨i (t)are obtained from the n×nidentity matrix by placing, respectively,
1 t 0 1
, 1 0
t 1
, and
t 0 0 t−1
in rows and columnsi,i+1.
Then B and B− are the subgroups of Gof upper- and lower-triangular matrices, respec- tively, and G≥0 is the subset of Gof matrices whose minors are all nonnegative.
The Weyl group is the symmetric group Sn. We let w0 := (i 7→ n+1−i) denote the longest permutation in Sn. For w ∈ Sn, we let ˙w ∈ G denote the signed permutation matrix which contains a ±1 in row w(k) and column k for each k ∈ [n], where the signs are chosen so that the submatrix with rows {w(1), . . . ,w(k)} and columns [k] has
determinant 1. For example,
˙ s1 =
0 −1 0
1 0 0
0 0 1
, and forw =s1s2, we have ˙w =
0 0 1 1 0 0 0 1 0
.
4.1 Unipotent radical U
nLet Un denote the subgroup of B of all unipotent matrices, andUn≥0 := G≥0∩Un its to- tally nonnegative part. We have the decomposition into totally positive Bruhat cells [26]
Un≥0= G
w∈Sn
Π>w0, where Π>w0 :=Un≥0∩B−wB˙ −.
The closure relation on cells is given by the strong Bruhat order.
We can take the link of the identity inUn≥0 to be the subset of matrices whose n−1 entries immediately above the diagonal sum to 1. It has a similar cell decomposition indexed by permutations w ∈ Sn with w 6= id. The natural inclusion Un ,→ G/B−
allows us to identify this link with one appearing in the proof ofTheorem 1.1for G/B−, which implies that it is a regular CW complex homeomorphic to a closed ball. This result was conjectured by Fomin and Shapiro [7] and proved by Hersh [19], all in general Lie type (cf.Corollary 1.3).
Example 4.1. Let n =3. Then
U3≥0 =
1 x y 0 1 z 0 0 1
: x,y,z,xz−y≥0
.
The permutation w0 = s1s2s1 = 321 indexes the cell where x,y,z,xz−y > 0, while w =s1s2=231 indexes the cell where x,y,z>0 andxz−y=0.
The link of the identity ofU3≥0 consists of all matrices inU3≥0with x+z=1. We can plot this region in the xy-plane:
x y
1 w0
s2s1 s1s2
s1
s2
.
We see that this agrees with the regular CW complex ofFigure 1.
4.2 Complete flag variety Fl
nTakingP=B, we can identifyG/Bwith the space Flnofcomplete flagsinRn, i.e. the space of tuples(V1, . . . ,Vn−1), whereVkis a subspace ofRn of dimensionk(1≤k≤n−1) and V1 ⊂ · · · ⊂Vn−1. (Here Vk is the subspace spanned by the first k columns of x ∈ G/B.) The totally nonnegative part Fl≥n0consists of all complete flags which can be represented by an element x ∈ Gwith nonnegative initial minors:
det(xI,[k]) ≥0 for all 1≤k ≤n−1 and I ∈ ([nk]). We have the decomposition (2.1) into totally positive Richardson cells
Fl≥n0 = G
v≤winSn
Π>(v,w0 ), where Π>(v,w0 ) :=Fl≥n0∩B−vB˙ ∩BwB.˙
The dimension ofΠ>(v,w0 ) is`(w)−`(v). The closure relation on cells is given by contain- ment of intervals in the strong Bruhat order:
(v,w) (v0,w0) ⇔ v0 ≤v≤w≤w0.
Example 4.2. Letn =3. Then Fl≥30gives a cell decomposition of a 3-dimensional ball, see Figure 2(left). Let us illustrate the homeomorphism (3.1) forg := (s1,s2s1). HereΠ>g0is an open line segment, and Star≥g0consists of 4 cells: a line segmentΠ>g0=Π>(s0
1,s2s1), two open square faces Π>(s0
1,w0) and Π>(id,s0
2s1), and an open 3-dimensional ball Π>(id,w0
0). This union is indeed homeomorphic to Π>g0×Cone(Lk≥g0) shown in Figure 2 (right). Here Lk≥g0 is a closed line segment whose endpoints are Lk>g,0(s
1,w0) and Lk>g,0(id,s
2s1), and whose interior is Lk>g,(0id,w
0).
4.3 Grassmannian Gr ( k, n )
Fix 1 ≤ k ≤ n−1, and take P to be the subset of G of all matrices whose lower-left (n−k)×k block is zero. Then we can identify G/P with the Grassmannian Gr(k,n), i.e. the space of k-dimensional subspaces of Rn. (Here x ∈ G/P corresponds to the k- dimensional subspace spanned by the firstkcolumns of x.) The totally nonnegative part Gr≥0(k,n) consists of all subspaces which can be represented by an n×k matrix whose k×kminors (known asPlücker coordinates) are all nonnegative.
In the case of Gr≥0(k,n), we can describe the decomposition (2.1) in terms ofpositroid cells [29]. Namely, let Bound(k,n) denote the set of bounded affine permutations, i.e. bijec- tions f :Z→Zsatisfying
f(i+n) = f(i) +nand i ≤ f(i) ≤i+nfor all i ∈Z, and
∑
n i=1(f(i)−i) =kn.
We write f in window notation as [f(1), . . . , f(n)]. Given an element of Gr(k,n) repre- sented by an n×k matrix M, we associate an element f ∈ Bound(k,n) as follows: for each i ∈ [n], we set f(i) ≥ i to be minimum such that row f(i) of M is in the span of rows i,i+1, . . . , f(i)−1 (where indices are taken modulo n). Then the positroid cell Π>f 0is defined to be the set of all elements of Gr≥0(k,n) associated to f, and
Gr≥0(k,n) = G
f∈Bound(k,n)
Π>f0.
The closure relation on cells is given by the dual of the Bruhat order on Bound(k,n). Example 4.3. Let (k,n) := (2, 4), and take g := [2, 4, 5, 7] ∈ Bound(2, 4). Then Π>g0
consists of all elements which can be represented by a matrix of the form
1 0
x1 0 x3 x4
0 1
with x1,x3,x4>0.
Now let us describe the map ¯νg from (3.1) and the dilation action ϑg. We have Star≥g0 = Π>g0tΠ>[3,4,5,60 ], and Lk≥g0 is a point, so Cone(Lk≥g0) 'R.
First we must fix a subset I ∈ ([42])whose Plücker coordinate does not vanish onΠ>g0. Here we may take any I 6= {1, 2}; let us take I := {1, 4}. This allows us to define the embedding ¯ϕI of an open dense subset of Gr(2, 4) into the affine flag variety:
ϕ¯I
1 0
x1 x2
x3 x4
0 1
=
1
z 0 0 0
−x1 1 0 xz2
−x3 0 1 xz4 0 0 0 1z
, where z is the formal loop parameter.
After performing some calculations in the affine flag variety and pulling back the result to Gr(2, 4), we find that
¯ νg
1 0
x1 x2 x3 x4
0 1
=
1 0
x1x4−x2x3
x4 0
x3 x4
0 1
,x2
x4
.
The first component gives a projection Star≥g0 →Π>g0. The dilation actionϑgis given by
t·
1 0
x1 x2
x3 x4
0 1
7→
1 0
x1+ (t−1)xx2x3
4 tx2
x3 x4
0 1
fort>0.
Acknowledgements
We thank Sergey Fomin, Patricia Hersh, Alex Postnikov, and Lauren Williams for stim- ulating discussions. We are also grateful to George Lusztig and Konni Rietsch for their comments on the first version of [12].
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